An Introduction to Engineering Fluid Mechanics
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1 An Introduction to Engineering Fluid Mechanics
2 Other Macmillan titles of related interest Jonas M. K. Dake: Essentials of Engineering Hydrology L. Huisman: Groundwater Recovery L. M. Milne-Thomson: Theoretical Hydrodynamics, Fifth Edition A. M. Muir Wood: Coastal Hydraulics John Pickford: Analysis of Surge R. H. J. Sellin: Flow in Channels A. Verruijt: Theory of Groundwater Flow E. M. Wilson: Engineering Hydrology M. Selim Valin: Theory of Hydraulic Models
3 An Introduction to Engineering Fluid Mechanics 1. A. FOX Departmen t of Civil Engineering University of Leeds Macmillan Education
4 ISBN ISBN (ebook) DOI / J. A. Fox 1974 Reprint of the original edition 1990 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission First published 1974 by THE MACMILLAN PRESS LTD London and Basingstoke Associated companies in New York Dublin Melbourne Johannesburg and Madras SBN
5 Contents Preface List of Principal Symbols IX xi I Definitions and Hydrostatics Basic definitions 1.2 Viscosity Non-newtonian fluids Specific mass, weight and gravity Pressure at a point in a fluid Pressure distribution in the atmosphere Hy'drostatic pressures in incompressible fluids Force on an inclined plane lamina Forces acting on curved surfaces Surface tension Manometry 16 2 Hydrodynamics The continuity equation The Euler equation Normal strain and deformation of a fluid element Rotation of a fluid element The Navier-Stokes equations The velocity potential The stream function Circulation Vorticity The source The sink The doublet The vortex The uniform wind Combinations of flow patterns Pressure distribution around a cylinder in uniform flow Forces acting on a cylinder The development of transverse forces The wake 68
6 Contents 2.20 Pressure distribution over an aerofoil The graphical addition of stream functions and velocity potentials The flow net Percolating flows 76 3 Dimensional Analysis Buckingham 1T theorems Construction of 1T groups The physical significance of some commonly used groups Models Examples of dimensional analysis Units Dimensional homogeneity of equations The Basic Equations of Engineering 110 Fluid Mechanics 4.1 The continuity equation The force equation The energy equation Flow through small orifices The venturimeter Notches Pipe diaphragm orifices The Pitot tube Applications of the force equation The variation of the Bernoulli constant across stream lines The free vortex Radial flow The free spiral vortex The forced vortex The Rankine vortex Vorticity 149 Boundary Layer Theory Formation of boundary layers The Prandtl mixing-length hypothesis Boundary layer separation Drag on spheres Secondary flow 168
7 Contents 6 Pipe Flow Simple experiments Laminar flow Turbulent flow Thefnumber S The Prandtl mixing length hypothesis applied to pipe flow The velocity distribution in smooth pipes The velocity distribution in rough pipes The universal pipe friction laws Losses in pipelines other than those due to pipe friction The energy grade line and the hydraulic grade line The energy coefficient The momentum coefficient Flow in pipe networks Analysis of pipe networks Open Channel Hydraulics Uniform flow Formulae for the Chezy C The Prandtl mixing length hypothesis applied to uniform free 234 surface flows 7.4 'Economic' channels S Flow in circular culverts and pipes Gradually varied non uniform flow in channels The analysis of gradually varied flow The specific force equation The specific energy equation Flow profiles The hydraulic jump The venturi flume Broad crested weirs and bed humps The sluice S The prediction of flow profiles in channels Method of integrating the gradually varied flow differential 268 equation Surge waves in channels 8 Pressure Transients Rigid pipe theory of waterhammer Sudden valve opening at the end of a pipeline Slow uniform valve closure 287
8 Contents 8.4 Elastic pipe theory Pressure surge caused by instantaneous valve closure Surge Tanks The frictionless analysis The frictional analysis Complex surge tanks Surge tank modelling Rotodynamic Machines Flow through rotating curved passages The reaction turbine Impulse turbines Centrifugal pumps Types of centrifugal pump The dimensional analysis of rotodynamic machines Unit speed, quantity and power The specific speed Scaling of results from model tests Cavitation 366 Further Reading 379 Index 381
9 Preface This book is for undergraduates and HNCjHND students in both civil and mechanical engineering. The accent throughout has been placed upon the engineering aspects of the subject but it is hoped that the more mathematically minded reader will find sufficient to interest him. Assumptions upon which analyses are based have been carefully specified. Any analysis is only as accurate as its underlying assumptions and so the reader should develop the habit of assessing the value of a piece of theory by considering the applicability of its assumptions in the context of the problem under examination. Both engineers and mathematicians have contributed to the study of fluid mechanics and of recent years there has been a marked tendency to use mathematical methods in place of the empiricism that was used in the past. I believe that this trend will continue and academic courses will become progressively more mathematical in their approach. The systems of units that have been used are the British system and the SI system. Even after the SI system has been completely introduced in the UK and Europe, the British system will still be used in many areas of the world. It will therefore be necessary for British engineers designing projects in these areas to know both systems. At the end of each chapter questions have been included which it is hoped will be of assistance in understanding the chapter. They are set in both systems of units, the SI values being enclosed in square brackets. Some questions come from examination papers of the University of London, the University of Leeds and the Part II hydraulics examinations of the Institution of Civil Engineers, and I gratefully acknowledge permission to use them; others have been evolved for this book. The answers supplied are of course my own. The subject is very large and it is not possible to cover every topic in detail. The student will need to read further and a list of suggested reading is included. I would like to thank Mr J. Higgins, of the Faculty of Applied Science, the University of Leeds, who prepared the drawings. Department of Ovil Engineering, University of Leeds J. A. Fox
10 list of Principal Symbols As far as possible all symbols used have been defined in the text as they occur, so any ambiguities arising out of the use of the same symbol to denote different variables can be easily resolved by reference to the text. The dimensions are given in parentheses. a and A A b andb b B (3 corc c Cd CD Cy C c C1 C d F area of flow (L 2 ) constant in the pump characteristic equation (Chapter 10) (L T2) angle (dimensionless) constant in the equation that describes the variation of j.i with temperature (0 -I, where 0 is the dimension of temperature) energy coefficient (dimensionless) breadth of a channel (L) surface breadth of a channel (L) breadth of a lamina (L) constant in the pump characteristlc equation (Chapter 10) breadth of a runner or impeller in the axial direction (L) the momentum coefficient (dimensionless) constant in the equation that describes how j.i varies with temperature (0-2 ) exit angle of a moving blade in a rotadynamic machirle (dimensionless) general constant wave velocity (Chapters 6, 8 and 9) (LT-1 ) coefficient of discharge of an orifice, notch or weir (dimensionless) coefficient of drag (dimensionless) coefficient of velocity of an orifice (dimensionless) coefficient of contraction of an orifice (dimensionless) coefficient of lift (dimensionless) Chezy constant in free-surface flows (L 112 T-1 ) constant in the pump equation (Chapter 10) (T2L-s ) depth of a flow (L) pipe diameter (L) height of opening below a sluice gate (Chapter 7) (L) critical depth of a free-surface flow (L) boundary layer thickness (L) laminar sublayer thickness (L) boundary layer displacement thickness (L) mean depth of a flow (Chapter 7) (L) small increment or decrement of a variable a finite difference specific energy (energy of flow referred to bed of flow) (L) Young's modulus of the pipe wall material (Chapter 8) (ML -I T-2 ) ideal or hydraulic efficiency (dimensionless) overall efficiency (dimensionless) mechanical efficiency (dimensionless) exponential constant ( ) (dimensionless) fractional valve opening (dimensionless) Froude number (dimensionless) frequency of a vortex trail (T-1 ) Darcy friction coefficient (T-1 ) force (MLr2) strength of a doublet (L 2T-1) velocity potential (L 2 T-1)
11 xii g r 'Y h j k L A m M IJ n v n p p 7'C 7T Q r R Re p s a List of Principal Symbols acceleration due to gravity (LT-2 ) circulation around a fluid boundary (L2T-1) angle of shear deformation in a solid (dimensionless) height of a rectangular lamina (L) potential head (pressure head plus height above datum) (L) height of a bed hump (L) energy loss/unit weight due to friction (L) inertia head (L) level difference between the two menisci of a manometer (L) total energy/unit of fluid referred to a horizontal datum (L) horizontal component of force acting on a lamina (MLT-2) static head (L) second moment of area (L 4) bed slope of a channel (dimensionless) hydraulic gradient (dimensionless) energy loss/unit weight/unit length of a channel flow (dimensionless) constant roughness number, Manning's, Kutta's or Strickler's (dimensionless) radius of gyration (L) bulk modulus of a fluid (ML -1 T-2) length of pipe (L) distance along a channel (L) lift force (MLr2) kineticity of a flow (Chapter 7) (dimensionless) hydraulic mean radius (in pipe flow) (L) hydraulic mean depth (in free-surface flows) (L) total momentum flow rate (MLT-1 ) coefficient of dynamic viscosity (ML -1 T-1) index (dimension variable) roughness number rotational speed of a pump or turbine (T- 1 ) specific speed of a pump or turbine ~dimensionless) coefficient of kinematic viscosity (L T-1 ) angular velocity of a forced vortex, a rotodynamic machine or a fluid element (T-l) pressure (ML -1 T-2) force (MLT-2) wetted perimeter of a flow channel (L) total force acting on a curved lamina (MLT-2) constant ( ) (dimensionless) dimensionless groups stream function (L2T-1) flow per unit width of a rectangular channel (L2T-1 ) flow leaving per unit length of a pipe (L 2 T-1 ) flow entering or leaving a channel per unit length (Chapter 7) (L 2 T-1 ) flow (L3rl) radius (L) radius (L) area ratio (Chapter 9) (dimensionless) Reynolds number (dimensionless) mass density (ML -3) pipe characteristic (Chapter 8) (ML -3) specific gravity of a fluid (dimensionless) distance (L) slope of sides of a channel (dimensionless) specific force of a free surface flow (L 2) coefficient of surface tension (MT-2 ) Poisson's ratio (Chapter 8) (dimensionless)
12 T T TO (J U u' U U v v' ii V w' w W x X X y y z List of Principal Symbols time (T) thickness (L) pipe period (T) viscous shear stress (ML -I T- 2 ) viscous shear at a boundary (ML -I T-2 ) angle (dimensionless) local velocity (L T-1 ) velocity component in the x direction (L T- 1 ) peripheral velocity of a rotating blade (LT- 1 ) -I time varying component of velocity in x direction in turbulent flow (LT ) steady component of velocity in x direction in turbulent flow (LT- 1 ) undisturbed stream velocity (L T- 1 ) stream velocity (L T- 1 ) velocity component in the y direction (L T- 1 ) time varying velocity component in the y direction in a turbulent flow (L T-1 ) mean velocity of a flow (L T- 1 ) velocity of a fluid particle on the centreline of a pipe velocity (L T-1 ) vertical component of a force acting upon a lamina (MLT-2 ) shear velocity (L T-1 ) velocity of a surge wave in a free-surface flow (LT-1 ) velocity of whirl (Chapter 10) (LT-I) velocity of flow (Chapter 10) (Lrl) relative velocity (Chapter 10) (LT-1 ) specific weight of fluid (ML -2T-2 ) component of velocity in the z direction (ML T- 1 ) time varying velocity component in z direction in turbulent flow (L T- 1 ) steady velocity component in z direction in turbulent flow (L T-1 ) weight of a fluid body (ML r2) weight flow (ML T-3 ) distance (L) body force acting in the x direction (ML T-2 ) distance along a lamina from its point of intersection with the free surface to the centroid (L) distance, usually in the direction of the earth's gravity field (L) distance from a pipe wall (L) body force acting in the y direction (MLT-2 ) elevation of a point above a datum (L) distance down from the surface of an area element (L) depth below the centroid of the cross sectional area of a flow (L) vorticity (T-1 ) xiii
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